Expression for Angular Frequency

In summary, the expression for angular frequency in an L.C.R. circuit derived from w = (1/LC - R2/4L2)1/2 is a determinant of Lx2 + Rx + 1/C = 0, and can be solved for w in terms of t by taking I as a function of t.This expression for angular frequency is found in a problem, and is related to the kirchhoff's loop rule. Additionally, the expression does not contain any w, and gives I as a function of t. The standard method for solving differential equations involving just dy/dx is to use the equation D = d/dt.
  • #1
zorro
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How is this expression for angular frequency in an L.C.R. circuit derived-

w = (1/LC - R2/4L2)1/2

(I came across this while solving a problem)
 
Last edited:
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  • #2
You solve the differential equation for kirchhoffs loop rule.
 
  • #3
Hi Abdul! :smile:
Abdul Quadeer said:
How does this expression for angular frequency in an L.C.R. circuit derived-

w = (1/LC - R2/4L2)1/2

(I came across this while solving a problem)

Well, without knowing anything about circuits, we can guess that it probably has something to do with the fact that that obviously looks like a determinant,

of Lx2 + Rx + 1/C = 0 …​

can you get a differential equation that looks like that, maybe for current? :wink:
 
  • #4
I got the same equation for w in place of x. On solving,

w = (-R +/- (R2 -4L/C)1/2)/2L

I did not get any differential equation.
 
  • #5
Abdul Quadeer said:
I got the same equation for w in place of x. On solving,

w = (-R +/- (R2 -4L/C)1/2)/2L

Are you sure? that's not the same ω as in your original post …
Abdul Quadeer said:
How is this expression for angular frequency in an L.C.R. circuit derived-

w = (1/LC - R2/4L2)1/2

I suspect that the R/2L is a separate exponential part

(my differential equation in I was from combining CVC = Q (so CdVC/dt = I), VR = IR, VL = LdI/dt, and d/dt(VC + VR + VL) = 0)
 
  • #6
tiny-tim said:
(my differential equation in I was from combining CVC = Q (so CdVC/dt = I), VR = IR, VL = LdI/dt, and d/dt(VC + VR + VL) = 0)

That will give a second order differential equation, right?
Your expression does not contain any w.
 
  • #7
It gives I as a function of t …

part of that solution will be harmonic, and ω will be the frequency of the harmonic part. :smile:
 
  • #8
I got
I/C + RdI/dt + Ld2I/dt2 = 0

How to solve this?
 
  • #9
Hi Abdul! :smile:

(just got up :zzz: …)
Abdul Quadeer said:
I got
I/C + RdI/dt + Ld2I/dt2 = 0

How to solve this?

Don't you know the standard method for these equations? …

write D = d/dt, so the equation becomes

(LD2 + RD + 1/C)I = 0,

which you can factor to

(D - R/2L + i√(1/LC + R2/4L2))(D - R/2L - i√(1/LC + R2/4L2))I = 0 …

carry on from there :smile:
 
  • #10
Its d2I/dt2 not (dI/dt)2 :wink:
 
  • #11
That's ok, that is what D2 means …

D2(I) = D(D(I)) = d/dt(d/dt(I)) = d/dt(dI/dt) = d2I/dt2 :smile:
 
  • #12


I have solved questions in D.E. like -
1 + (dy/dx)2 = xdy/dx

by taking dy/dx = p

Do we use p2 for both (dy/dx)2 and d2y/dx2 ?
 
  • #13
no …

that p equation has an xdy/dx,

the D is for equations with just dy/dx :smile:
 

1. What is the formula for angular frequency?

The formula for angular frequency is ω = 2πf, where ω is the angular frequency in radians per second and f is the frequency in hertz.

2. How is angular frequency related to linear frequency?

Angular frequency is directly proportional to linear frequency. This means that as the linear frequency increases, the angular frequency also increases proportionally.

3. What is the unit of measurement for angular frequency?

The unit of measurement for angular frequency is radians per second (rad/s).

4. How is angular frequency used in physics?

Angular frequency is used in physics to describe the rate at which an object rotates or oscillates about a fixed axis. It is also used in calculations involving circular motion, such as centripetal force and torque.

5. Can angular frequency be negative?

Yes, angular frequency can be negative. A negative angular frequency indicates that the object is rotating in the opposite direction of positive angular frequency. This is often seen in phenomena such as retrograde motion in astronomy.

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